163. In every arithmetical progression, having a finite number of terms, there are five quantities especially to be considered, viz.: The first term, the last term, the number of terms, the common difference, and the sum of the terms. When any three of these are given, the other two may be found. In investigating rules for the solution of these different cases, let us denote

The first and last terms are called extremes, all the other terms are called Arithmetical means.

Given a, d, and n, to find 7.

164. The second term is, by definition, equal to ad; the third is equal to the second, increased by d, that is, it is a +2d; the fourth term is equal to the third, increased by d, that is, it is a + 3d; and on : hence, the nth term, or, is equal to a + (n-1)d; or,

l = a + (n − 1)d

(1.)

That is, any term is equal to the first term, plus the product of the common difference by the number of preceding terms.

EXAMPLES.

1. The first term is 3, and the common difference is 3. What is the 7th term.

Here, the number of terms preceding the 7th, is 6; hence, by the rule, the 7th term is, 3+3 x 6, or 21

2. The first

-3; what is

term is 24, and the common difference the 5th term?

3. The first term is 1, and the common difference is 1; what is the 25th term?

165. Having given a, d, and n, l may be found by the preceding rule. Having found the nth term, l, the preceding term is equal to -d; the term preceding that, 2d, and so on. If now, we write out the terms of the progression, and then write the same terms in a reverse order, the sum will be the same in both cases.

Hence, we have,

8 = a + (a+d) + (a+2d) + ... + (l−2d)+(l−d) +l s = 1 + (l−d) + (l−2d) + ... (a+2d) + (a+d)+a. Adding these equations, term by term, we have, 28 = (a+1)+(a+1)+(a+1)+...+(a+1)+(a· +1)+(a+l). Here, (a+1) is taken n times; hence,

28 = n(a+1); or, sin(a+l) . . (2.)

That is, the sum of the terms is equal to the sum of the extremes, multiplied by half the number of terms.

Formula (2) can be placed under another form, by substituting for 7 its value taken from Formula (1).

by means of which, the sum of the terms may be found more directly than by Formula (2).

EXAMPLES.

1. The first term is 2, the common difference is 3, and the number of terms is 17.

What is their sum?

and the number of terms

2. The first term is, the common difference

20. What is the sum?

3. The first term is 20, the common difference is 2, and the number of terms is 6. What is the sum?

4. The first term is 5, the common difference 3,

and the number of terms

12.

What is the sum?

Ans. 258.

5. The first term is 2, the common difference is

3, and the number of terms is 10. sum?

Formulas (1) and (2), contain five quantities: a, 17, n, l, and s. If any three be assumed at pleasure, the remaining two may be deduced from the formulas.

166 A GEOMETRICAL PROGRESSION is a series, cach term of which is derived from the preceding one, by multiplying it by a fixed quantity. This quantity is called the ratio of the progression.

167. When the first term is positive, and the ratio greater than 1, each term is greater than the preceding one, and the progression is said to be increasing. When the ratio is less than 1, each term is less than the preceding one, and the progression is said to be decreasing.

Thus, the series 2, 4, 8, 16, &c., is an increasing progression, whose ratio is 2.

The series 2, 1, 4, 4, &c., is a decreasing progression, whose ratio is 1.

168. When the ratio is negative, the terms of the progression are alternately positive and negative. The positive terms make up a progression whose ratio is equal to the square of the given ratio, and the nega tive terms make up a second progression, having the same ratio.

The progression 2, 4, 8, 16, &c., whose ratio is 2, is made up of the two progressions, 2, 8, 32, &c., and 4, — 16, -64, &c., whose ratio in each case is 4.

169. In any geometrical progression, there are five quantities considered, any three of which being given, the other two may be found These quantities are:

The first and last terms are called extremes; all the other terms are called Geometrical means.

Given a, r, and n, to find 7.

170. The second term is, by definition, equal to the first multiplied by r, that is, it is equal to ar; the third term is equal to the second, multiplied by r, that is, it is equal to ar2; the fourth term is equal to the third, multiplied by r, that is, it is equal to ar3; and so on to the nth term, which is equal to ɑrn. ; hence,

1 = arn -1

1

(1.)

That is, any term of a geometrical progression is equal to the first term, multiplied by that power of the ratio whose exponent is equal to the number of preceding

terms.

EXAMPLES.

1. Find the 7th term of the series 1, 4, 16, &c.

We have,

1 = arn−1 = 1 × 46 = 4096. Ans.

2. Find the 8th term of the series 2, 4, 8, &c.

Ans. 256.

8. Find the 12th term of the series 30, 15, 71⁄2, &c.

Ans.

4. Find the 8th term of the scries 5, 25, 125, &c.

Ans. 390625